DocumentCode :
3287403
Title :
Linear feedback control of a von Kármán street by cylinder rotation
Author :
Borggaard, J. ; Stoyanov, M. ; Zietsman, L.
Author_Institution :
Interdiscipl. Center for Appl. Math., Virginia Tech, Blacksburg, VA, USA
fYear :
2010
fDate :
June 30 2010-July 2 2010
Firstpage :
5674
Lastpage :
5681
Abstract :
This paper considers the problem of controlling a von Kármán vortex street (periodic shedding) behind a circular cylinder using cylinder rotation as the actuation. The approach is to linearize the Navier-Stokes equations about the desired (unstable) steady-state flow and design the control for the regulator problem using distributed parameter control theory. The Oseen equations are discretized using finite element methods and the resulting LQR control problem requires the solution to algebraic Riccati equations with very high rank. The feedback gains are computed using model reduction in a “control-then-reduce” framework. Model reduction is used to efficiently solve both Chandrasekhar and Lyapunov equations. The reduced Chandrasekhar equations are used to produce a stable initial guess for a Kleinman-Newton iteration. The high-rank Lyapunov equations associated with Kleinman-Newton iterations are solved by applying a novel model reduction strategy. This “control-then-reduce” methodology has a significant computational cost, but does not suffer many of the “reduce-then-control” setbacks, such as ensuring the unknown feedback functional gains are well represented in the reduced-basis. Numerical results for a 2-D cylinder wake problem at a Reynolds number of 100 demonstrate that this approach works when perturbations from the steady-state solution are small enough. When this feedback control is applied to a flow where vortex shedding has already occurred, the feedback control in the nonlinear problem stabilizes a nontrivial limit cycle. This limit cycle does have reduced lift forces and showcases the promise of the linear feedback control approach.
Keywords :
Lyapunov methods; Navier-Stokes equations; Newton method; finite element analysis; flow control; vortices; wakes; 2D cylinder wake problem; Kleinman-Newton iteration; LQR control problem; Navier-Stokes equations; Oseen equations; Reynolds number; actuation; algebraic Riccati equations; circular cylinder; control-then-reduce framework; cylinder rotation; distributed parameter control theory; feedback functional gains; finite element methods; high-rank Lyapunov equations; lift forces; linear feedback control approach; nonlinear problem; nontrivial limit cycle; novel model reduction strategy; perturbations; reduced Chandrasekhar equations; regulator problem; steady-state flow; steady-state solution; von Karman vortex street; vortex shedding; Computational efficiency; Control theory; Feedback control; Finite element methods; Limit-cycles; Navier-Stokes equations; Reduced order systems; Regulators; Riccati equations; Steady-state;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
American Control Conference (ACC), 2010
Conference_Location :
Baltimore, MD
ISSN :
0743-1619
Print_ISBN :
978-1-4244-7426-4
Type :
conf
DOI :
10.1109/ACC.2010.5531133
Filename :
5531133
Link To Document :
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